Topics
6. Equations & Inequalities
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United States of America
Grade 7

6.08 Problem solving with inequalities

Lesson
Practice

Introduction

We have now looked at solving inequalities that involve  one step  or  two steps  to solve. We're now going to take a look at how we can use inequalities to solve problems given a real world problem.

Solve inequality problems

Much like  solving equations  from real world problems, there are certain keywords or phrases to look out for. When it comes to inequalities, we now have a few extra keywords and phrases to represent the different inequality symbols.

Keywords:

  • > greater than, more than.
  • \geq greater than or equal to, at least, no less than.
  • < less than.
  • \leq less than or equal to, at most, no more than.

When working to solve problems always be sure to check your answer to see if it is reasonable and then check it by substituting into your inequality.

Examples

Example 1

Construct and solve an inequality for the following situation:

"The sum of 2 groups of x and 1 is at least 7."

Worked Solution
Create a strategy

Translate the phrases into mathematical symbols then solve the inequality by isolating the x on side of the inequality.

Apply the idea

The phrase "at least" means the same as "greater than or equal to", "groups of" means multiplication, and "sum" means addition.

\displaystyle 2x+1\displaystyle \geq\displaystyle 7Write the inequalty
\displaystyle 2x+1-1\displaystyle \geq\displaystyle 7-1Subtract 1 from both sides
\displaystyle 2x\displaystyle \geq\displaystyle 6Simplify
\displaystyle \dfrac{2x}{2}\displaystyle \geq\displaystyle \dfrac{6}{3}Divide both sides by 2
\displaystyle x\displaystyle \geq\displaystyle 3Simplify

Using a number line, the solution to this inequality is:

-2-1012345678910

So the possible values of x are those that are greater than or equal to 3.

Example 2

Lachlan is planning on going on vacation. He has saved \$2118.40, and spends \$488.30 on his airplane ticket.

a

Let x represent the amount of money Lachlan spends on the rest of his holiday. Write an inequality to represent the situation, and then solve for x.

Worked Solution
Create a strategy

The amount of money that he can spend on his holiday is up to but not more than the difference between his savings and the amount spent on the airplane ticket. Translate this information into mathematical symbols.

Apply the idea

The phrase "up to but not more than" means that we are going to use the \leq symbol and the phrase "difference between" means subtraction.

\displaystyle x\displaystyle \leq\displaystyle 2118.40 - 488.30Write the inequality
\displaystyle x\displaystyle \leq\displaystyle \$1630.10Evaluate
b

What is the most that Lachlan could spend on the rest of his holiday?

Worked Solution
Create a strategy

Determine the largest value of the inequality from part (a) by recalling the definition of inequality symbol.

Apply the idea

The inequality x \leq\$1\,630.10 means that x can take any value that is less than or equal to \$1\,630.10. So the maximum amount that Lachlan can spend on his holiday is \$1\,630.10.

Example 3

At a sport clubhouse the coach wants to rope off a rectangular area that is adjacent to the building. He uses the length of the building as one side of the area, which measures 26 meters. He has at most 42 meters of rope available to use.

If the width of the roped area is W, form an inequality and solve for the range of possible widths.

Worked Solution
Create a strategy

Translate the given information into mathematical symbols and use the perimeter equation to solve for the possible width of the rope.

Apply the idea

One side of the roped area is the length of the clubhouse. So the situation looks something like this:

A layout of the clubhouse and the roped section. Ask your teacher for more information.

The length of the roped area is equal to the length of the clubhouse which is 26 meters.

\displaystyle \text{width + width + length} \displaystyle =\displaystyle \text{Perimeter}Rope perimeter is equal to the sum of two widths and the length
\displaystyle 2W+26\displaystyle \leq\displaystyle 42Substitute the values and variables
\displaystyle 2W + 26 - 26\displaystyle <\displaystyle 42 - 26Subtract 26 from both sides
\displaystyle 2W\displaystyle \leq\displaystyle 16Simplify
\displaystyle \dfrac{2W}{2}\displaystyle \leq\displaystyle \dfrac{16}{2}Divide both sides by 2
\displaystyle W\displaystyle \leq\displaystyle 8\text{ meters}Simplify

The maximum width should be 8 meters to be able to use all of the 42-meter rope and we will have enough.

Idea summary

Keywords and phrases to represent the different inequality symbols:

  • > greater than, more than.
  • \geq greater than or equal to, at least, no less than.
  • < less than.
  • \leq less than or equal to, at most, no more than.

Outcomes

7.EE.B.4

Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

7.EE.B.4.B

Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

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